Model description

Antibodies develop in domains within secondary lymphoid organs called germinal centers (GCs), which appear shortly after infection [12]. B cells with a moderate threshold affinity for the antigen (Ag) are activated and seed GCs. These B cells then undergo an evolutionary process of mutation and selection that results in B cells with higher affinity receptors as time ensues [13]. The AID gene introduces mutations into the B cell receptor (BCR) at a high rate in GCs. The mutated B cells undergo selection against Ag, which is displayed on Follicular dendritic cells (FDC) on immune complexes (IC) [14] (Fig 1a). The B cells attempt to capture Ag by forming transient synapses with the FDCs [15]. Captured Ag is processed and presented on their surface as peptide-MHC class II complexes. The B cells then compete with each other to bind to T follicular help cells (TfhCs) via interactions between these peptide-MHC complexes and the T cell receptor on the surface of TfhCs. Successful binding results in a survival/proliferation signal. B cells that display more peptide-MHC complexes have an advantage in this competition[16]. The majority of positively selected B cells undergo further rounds of mutation and selection [17–19]. Some of the positively selected B cells differentiate into antibody-secreting plasma cells and memory cells. As time progresses, antibodies with increasing affinity for the Ag are generated.

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Fig 1. Extraction of Ag molecules from an immune complex on an antigen-presenting cell.

(a) A B cell approaches a FDC in order to capture antigen (Ag). The FDC presents an immune complex holding a virus (purple) whose surface is covered by spike proteins (yellow), which are the Ags. The attachment of the virus to the FDC is mediated by low-affinity antibodies (green) bound to receptors, called Fc Receptors (orange, FcR), on the FDC surface. (b) Schematic representation of the synapse between the B cell and the FDC (top view). The apparent virus surface is of radius R, where Ag (spike) molecules are distributed. The BCR (black) scans a region, attempting to capture Ag. (c) The extraction of antigen molecules from a FDC by a B cell: (c-i) A protrusion containing the BCR extends to touch the surface of the FDC displaying Ag molecules. (c-ii) Depending on the Ag concentration on the FDC, the BCR may encounter Ag molecules to which it can attach. (c-iii) Once a BCR-Ag bond has formed, the B cell pulls on it by applying force mediated by actin. (c-iv) Extraction of Ag from FDC can occur. (d) Three possible outcomes to the extraction attempt: The initial bond between the BCR and the Ag is severed and no Ag is extracted; the BCR bond holds while the IC-Ag bond breaks and one Ag molecule is extracted; the second arm finds another Ag before the first IC-Ag bond breaks and two Ag molecules are extracted. A scenario where the entire virus+Ab is extracted from the FDC is also feasible and gives similar results in our model.

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Affinity maturation has been studied extensively using theoretical models over the last decades, mostly using population dynamics approaches [19–24] and more recently, using detailed simulation of the dynamics of the immune cells invovled in the GCR [25]. To explore how B cell selection and the generation of high affinity Abs depend on Ag concentration and presentation, we constructed a simplified model of Ag capture from the FDCs, and a coarse-grained model of B cell selection in a GC. Thus, our model generalizes previous modeling approaches to include the interaction of the BCR with Ag (spike).

Antigen distribution on FDCs

We are interested in ICs presenting a virus, or a virus fragment (Fig 1a). The Ags of interest are the spike proteins distributed on the virus surface. Assuming that the spikes are distributed on the surface of the virus of radius R, with average density n_Ag, it contains a total of M_Ag = 4πR²n_Ag spikes (Fig 1b). A virion particle with a radius of 120nm (Table 1) with spike density of 0.35 spikes/100nm², has about M_Ag = 160 spikes on its surface. During the Ag capture process, BCRs scan for Ag over a region of the synapse and encounter an Ag molecule with probability p. We assume that the number N_Ag of Ag molecules in the scanning area is distributed randomly, according to the Binomial distribution (N_Ag ~ B(M_Ag, p)).

Ag encounter and internalization by the BCR

Consistent with data showing that B cell protrusions that extended toward FDCs retract if the associated BCRs do not find Ag [26], we assume that the BCR has a characteristic time to find the Ag (see Methods). If it does not bind to Ag during that time, no Ag is captured (see Fig 1ci and 1cii). Once one of the arms of the BCR binds to Ag, an Ag molecule may be pulled away by force[27]. At this stage, there is a tug-of-war over the Ag between the BCR and the IC (see Fig 1ciii). Characteristically, when the binding energy between the BCR and the Ag is much larger than the binding energy between the Ag and the virus/Fc receptor/FDC membrane (see Fig 1), the Ag will be captured by the BCR. We denote the potential interaction energy required to extract the Ag by E_Ag−mem, and the interaction energy of the BCR/Ab and Ag by E_Ag−Ab. A successful Ag capture event occurs when the bond between the Ag and the membrane ruptures before the bond between the BCR and the Ag (see Methods).

When the off-rate of the BCR arm from the Ag is smaller or of the order of the effective on-rate (qN_Ag), and if one arm is attached initially (see Fig 1cii), the second BCR arm can bind to another Ag molecule if one is available. Thus, a pulling attempt can have three possible outcomes, with zero, one or two Ag molecules captured (see Fig 1d), depending on the binding affinity and the number of available and accessible Ag molecules N_Ag.

Simplified model of GC reaction

In the first days following infection or immunization, M different clones (unique BCRs) of activated B cells proliferate with little competition, creating a pool of cells on which AM operates[28]. Few or no mutations are introduced to the BCR sequence at this early stage. We use a simple birth/death process to describe this initial growth stage.

B cells then start to mutate, and whether or not a B cell is subsequently positively selected depends on its ability to internalize Ag and compete with other B cells for survival signals by interacting with TfhCs. TfhCs have an important role in regulating the duration of the cell cycle in B cells during AM [17,18]. Following a TfhC signal, B cells divide (and mutate) multiple (4–6) times before going back for another round of selection [29–31].

Most theoretical models enforce selection by eliminating cells with low affinity BCR [23,24]. It has been shown [17] that TfhCs control the rate at which B cells go through the cell cycle. B cells that receive strong proliferation signal from the TfhCs divide multiple times in the dark zone before going back to the light zone. We model this behavior by using a birth-rate for B cells which is proportional to the amount of captured Ag.

The proliferation rate of B cells is a function of the number of captured Ag molecules, further modulated through competition for TfhCs. Indeed, it was shown [32], that TfhCs preferentially interact with B cells that have captured more Ag, presumably giving them a stronger proliferation signal. To mimic this competition we assume the birthrate of B cell i to be: (1) where 〈A〉 is the average amount of Ag consumed by all B cells that captured at least 1 Ag (B cells that do not capture Ag have zero birth rate); β₀ is the basal birthrate, and C is a measure of the availability of TfhC help. When C>>〈A〉, all B cells get the same amount of metabolic boost and divide at the same rate β₀. When C<<〈A〉 TfhC help is limiting and the cell birthrate is proportional to the amount of captured Ag. The GC has a limited capacity and grows during this competitive phase according to a stochastic logistic growth process. Thus, the selection process depends on the number of B cells in the system through the logistic death rate (Methods Eq (6)). This logistic term accounts for the increased competition between B cells to receive a survival signal from TfhCs when the number of the latter is limiting. TfhCs are essential for GC maturation of B cells, and without them, AM of B cell cannot occur [33]. We assume here that the number of TfhCs is not so small as to stop the formation of the GC. Thus, at the limit where C goes to zero, our model does not corresponds to the absence of TfhCs in the GC. However, we do use the reduction of C to model increased B cells competition in a GC when the number of TfhCs is gradually reduced, as occurs during the first stages of HIV infection. For these reasons, we do not consider here the limit of C→0. In a more complete proliferation model C should depend on the total number of B cell in the GC and thus change with time. However, the qualitative behavior of our model would not change if the parameter C would have such explicit dependency on the GC size as upon HIV infection we expect Tfh levels to be smaller at all times than upon infection with other pathogens.

Affinity change following BCR mutation

During AM, B cells mutate the BCR encoding gene using the AID enzyme. This results in changing cytosine to tyrosine on one DNA strand [34]. We modeled the effect of mutation as a change in the on-rate q or the interaction energy E_Ag−Ab upon cell division (see Methods). We convert the interaction energy to the off-rate by , and the affinity of the BCR is calculated as ω = q/r₀. Our simplified model assumes that affinity improvements and reductions (through q or E_Ag−Ab) are equally likely following mutation and that all mutations change affinity. These choices are certainly not realistic, but making advantageous mutations rarer than deleterious ones, or making some of the mutations silent or lethal (as in reality) do not change the qualitative results.

Highest affinity antibodies are generated at an optimal spike density

To explore the role of SD on AM, we calculated the affinity of the most dominant clone at the end of the GC reaction (GCR), which is day 16 of the competitive phase. The affinity of this clone will play an important role in the resulting humoral immune response. An important observation is that the affinity of the dominant clone varies non-monotonically with the SD (n_Ag). Fig 2a demonstrates this finding for a TfhC level that was limiting across the range of n_Ag (SD) spanning 0.01 to 0.36. There is also a pronounced asymmetry in the variation of affinity with n_Ag. As n_Ag increases from very small values, there is a sharp rise in affinity, while its drop for larger n_Ag is gradual. The number of B cells in the GC is not reduced when SD is high. Since B cells have to capture at least 1 Ag molecule in order to attempt proliferation, the average proliferation rate increases with SD.

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Fig 2. Affinity maturation (AM) in a germinal center (GC).

Simulations of the GC reaction starting with 20 different B cell clones. During AM, cells mutate their BCR upon division, changing only the BCR interaction energy with the Ag (E_Ag−Ab) and thus the off-rate r₀ of the BCR from the Ag. (a) The mean affinity (ω = q/r₀) of the dominant clone at day 16 of the GC reaction in the competitive phase, as a function of spike density n_Ag. (b) Mean number of Ag molecules captured by the dominant clone at day 16 as a function of the spike density. (c) The capture probability of Ag molecules by a single BCR (Eq (S10)) when the binding energy of the BCR is E_Ag−Ab = 1.5. The other parameters of the captures process are listed in Table 2. (d) The probability of selecting the weaker affinity B cell (see text) during the competitive phase. This probability depends on the spike density (Eq (S11)). We sampled energies of B cell pairs from the population energy distribution (S5 Fig) obtained in the simulation results shown in (a), to estimate the probability of the lower affinity cell proliferating before a higher affinity one, at different times during the GCR. The relative mild variation in this figure are amplified in multiple rounds of selection.

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These qualitative results are robust to variations in the other parameters in the model (S1 Fig), and for mutations resulting in changes in q (S2 Fig) or E_Ag−Ab (Fig 2a). (The variation of affinity with SD is less pronounced when mutations change q for technical reasons described in S2 Fig.)

In the GC, memory and plasma B cells are produced throughout the GCR [12]. As a proxy for the affinities of memory and plasma B cells produced during the maturation process, we randomly selected 10% of the B cells at each intermediate time point. The resulting average, displayed in (S3 Fig) still exhibits a non-monotonous dependence on SD similar to that observed in Fig 2a.

The mechanistic reason underlying these results relates to the amount of antigen that is internalized by B cells as n_Ag changes (Fig 2b). Upon increasing n_Ag from a small value, the amount of Ag internalized by B cells grows significantly as it becomes more probable to bind one or two Ag molecules (see Fig 2c). However, once the SD is sufficiently high, most BCRs bind/internalize on average the same amount of Ag irrespectively of their affinity (if it is above a threshold value), and it is difficult to distinguish B cells based on the affinity of their receptor for the antigen. Competition between B cells to be positively selected is not severe in the latter regime, limiting the driving force to increase affinity by further mutations. Thus, the affinity of the antibodies produced begins to decline beyond an optimal SD. The gradual drop-off in affinity as n_Ag becomes too high, compared to the sharp decline when n_Ag becomes too low, is because the number of internalized antigens rises very sharply as n_Ag increases from a low value, but then slowly saturates to its maximum for sufficiently high SDs as depicted in Fig 2b. In the limit of high n_Ag, there is a finite probability for the BCR to capture 1 or 2 Ag mols. At this limit, the probability to extract one Ag mol is given by r/(r + λ) and the probability to extract two Ag mols is λ/(r + λ), where λ is the off-rate of an Ag mol from the membrane when one arm of the BCR is bound and r is the rupture rate of the BCR from an Ag mol when one arm of the BCR is bound.

Our model suggests that clonal diversity also varies with SD. When SD is low, antigen is scarce and the selection forces will be fiercer. That could result in a more rapid loss of diversity. We estimate diversity by computing the fraction of the GC comprised of the dominant clone (S2b Fig, S4a Fig), which we denote by f_d. For very low SD a few dominant clones comprise most of the GC’s B cells at the end of the GCR (large f_d). This is because for low SD, selection during AM is dominated by the large competitive advantage, compared to most B cells, of a few B cells that internalize more antigen due to stochastic fluctuations. As the SD increases, a greater number of B cells can internalize antigen successfully and the GCR produces a more diverse clonal population.

The fraction of dominant clones has a large variability across different GC realizations. Indeed, it has been shown [31] that while some GCs appear to have been taken over by a single clone at day 16, others show high clonal diversity. This is the direct result of stochastic selection forces at play in the GC [35]. A similar behavior is observed in our GC model (see S4 Fig). Here, the source of stochasticity is the random Ag capture process, as well as the stochastic proliferation and death of B cells.

While the fraction of the dominant clone mostly decreases with increasing SD, the higher affinity of the BCR at intermediate density contributes to a small increase in f_d. Thus, f_d is not monotonically decreasing with SD. Indeed, the improved selection at an intermediate density leads to greater loss of diversity in this range, which appears as a small bump in f_d (approximately between n_Ag = [0.1, 0.18], see inset S4 Fig). This effect is more pronounced when the overall diversity loss is slower as a result of a smaller death rate (see S1 Fig).

Selection forces act with the highest fidelity at an intermediate spike density

To understand how cell selection in AM is related to Ag density, we estimated the probability P_{weaker aff selection} that, in any given round of mutation and selection, a B cell with a lower affinity for the antigen gets selected and expands in favor of another B cell that has a higher affinity. A B cell with a higher affinity receptor is likely to internalize more antigen, be more successful at obtaining a survival signal from T helper cells and proliferate more. However, because Ag capture is probabilistic, and the birthrate relates to the amount of captured Ag, there is a chance that a B cell with lower affinity will be selected in favor of one that expresses a higher affinity receptor.

To empirically estimate p_{weaker aff selection}, we sampled B cell pairs from the affinity distribution computed in our simulations (see S5 Fig). We detail how the probability is estimated in the methods section. Interestingly, we find that the probability changes with time, and depends on Ag density (Fig 2d). Importantly, p_{weaker aff selection} is minimal at intermediate Ag density, at a value similar to the maximum for the BCR affinities. Thus, very high or low SDs lead to poor differentiation between higher and lower affinity B cells. Because the B cells with a higher affinity are most likely to be chosen to proliferate at an intermediate SD, selection is strongest for such SD, and thus highest affinity antibodies evolve.

Cooperative binding with two arms of the BCR leads to optimal affinity increase at intermediate densities

To further elucidate the origin of the maximum in antibody affinity at an intermediate SD, we studied the time evolution of the mean affinity of the B cell population. We aimed to find the dependency of the optimal density on the rates of different processes in our model, rather than attempt to recapitulate our simulation results. Assuming that the time τ to find the first Ag molecule before retracting the B cell protrusion is very long and that each B cell has only a single BCR, we find a mean-field equation (see SI) for the evolution of the mean on-rate (2) where 〈A〉 is the average amount of captured Ag by the B cell population (here 〈A〉 is between 0 and 2), cov(q, A) is the covariance between the number of captured Ag molecules A and q, C₁ is the parameter that corresponds to TfhC availability parameter and τ₀ ∝ (β₀−μ₀)^-1 is the time scale of the process. We recall Fisher’s fundamental theorem of natural selection stating that “The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time” [36]. Eq (2) is a generalization of Fisher’s fundamental theorem and is also reminiscent of Price’s equation [37] if we consider q to be a trait of the population. We further find that the variance σ² of the on-rate distribution evolves as (3)

Solving Eqs (2) and (3) numerically (S6 Fig), we find that exhibits the same non-monotonic behavior as in our simulation results S2a Fig. When TfhC level is scarce (small C₁), the affinity increases faster due to the increased competition between B cells (S6 Fig). The position of the optimal density changes with time (S6 Fig) towards smaller Ag densities. For a given on-rate distribution (fixed mean and variance), the rate of increase in affinity is maximal at density n* given by (4) where λ and ξ are the off-rates of the Ag from the IC when one or two arms of the BCR are bound, respectively. r and k are the BCR arm rupture rate from the Ag when one or two arms of the BCR are bound to Ag, respectively (see SI).

When the interaction energy of the Ab and the Ag is equal to that of the Ag with the IC (E_Ag−Ab = E_Ag−mem), the optimal density is (5) where λ₀ is the disassociation rate of the Ag from the IC. Thus, at the optimal selection density, the characteristic effective on-rate () is equal to λ₀. At this density, B cells spilt into different populations, each capturing a different amount of Ag.

Interestingly, for very large values of q, the number of captured Ag molecules A is independent of q as the covariance between them goes to zero (see SI). In this limit, it is impossible to differentiate between B cells affinities and the mean affinity stops increasing, reaching an asymptotic value. Similarly, the variance of the distribution reaches an asymptotic value.

Finally, we speculated whether a scenario where the BCR has one arm (see S7 Fig) would still show the non-monotonic dependence on Ag density. We estimated the time evolution of for a hypothetical BCR with one arm (see SI and S7 Fig). Interestingly, when the time τ to find the first Ag molecule is of the same order as the effective on-rate (basal on-rate multiplied by the density), the mean affinity has a maximum at intermediate densities (see S7 Fig). However, τ is likely much larger than the on-rate, in which case the affinity is monotonically decreasing as a function of the Ag density (see S7 Fig). We conclude that the optimal SD observed in our simulations is a direct result of the cooperativity between the two arms of the BCR because the analytical model shows that an optimal SD emerges only when the BCR has two arms. The cooperativity allows a second arm to search and bind an Ag molecule while the first is bound to another Ag molecule on the surface of the IC.

Ag depletion improves AM at high SD

During AM, Ag is depleted from IC as it is being consumed by the B cells. As shown in the previous sections, AM depends on the ability of the GCR to differentiate bewteen B cells of different affinities. Optimal selection is achieved when only B cells with the highest affinity are likely to capture 2 Ag molecules (see Eq (5)). Depletion of Ag from FDCs during AM will result in fewer immune complexes encountered by the BCRs. It will not change the local SD on the virus. Rather, it will reduce the first encounter probability of Ag molecules by a BCR (See Eq (7)). To study how the competative forces may be modulated in time, we studied a hypothetical scenario where SD exponentially decays during AM (). Interestingly, when the SD decays, selection improves when the initial SD is high (S8 Fig). As a result, the affinity at day 16 of the GCR is higher compared to the fixed SD scenario (Fig 2a). At high SD, as the affinity of the B cell population improves, decreasing SD allows the GCR to remain at close to the optimal differention point. However, decreasing Ag density harms the development of GCs for which the initial SD is low. Since B cell have to capture at least one Ag mol to proliferate, most GCs do not succeed in reaching day 16 (S8 Fig) in this limit.

Depleted TfhC numbers during HIV infection affects AM in a way that favors a low virus SD

HIV dominantly infects CD4 T cells and significantly reduces their number immediately following infection during a time when the first antibody responses are developing in the host [38]. We therefore explored the effects of making TfhC activity even more limiting (than what is usual). To do so, we changed the parameter C (Eq (1)), which represents the availability of TfhCs during AM. Again, we use the affinity of the dominant clone at the end of the GCR as a metric of the affinity of antibodies generated. Previously (Fig 2a) we found that for high SDs, antibody affinity was lower than optimal. This was because, for sufficiently high SD, most BCRs internalized one or two antigens and so competition between B cells was restricted. But, when the level of TfhC (C) is even more limiting, even for high SDs, small differences between B cells with regard to the amount of internalized antigens are amplified by the intense competition between B cells for interacting with, and receiving a survival signal from TfhCs (Fig 3a, S9 Fig). Thus, if the SD is high, more potent antibodies are generated as TfhC levels decline. However, when the SD is low, decreasing C does not alter the affinity of the resulting antibodies significantly. Consistent with this result, the probability of selecting the lower affinity B cell decreases as C decreases (see Fig 3b). This is because selection is a stronger force when there are fewer TfhCs. In agreement with our result in Fig 3a, this enhancement in selection forces is more pronounced at intermediate and high SDs compared to low SDs. Thus, as TfhC levels are smaller, high affinity antibodies can be generated more readily for high SDs. Thus, perhaps, HIV, which is the only virus that principally targets T helper cells and depletes their numbers, evolved a low SD to prevent strong antibody responses from developing during the early stages of infection when T helper cell levels rapidly decline. (Note that Human T cell Leukemia viruses also infect T cells, but they do not result in a sharp decline in T helper cell numbers [39]. Their effect on the number and functioning of TfhCs is still unclear [40]).

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Fig 3. Effect of T helper cell level and spike density on affinity maturation.

(a) The average affinity of the dominant clone in a GC increases for viruses exhibiting a high spike density as Tfh levels (C) decline. Different curves correspond to different values of spike density (n_Ag). E_Ag−Ab changes upon mutation in these simulations. (b) The probability of selecting the wrong B cell (see text) is shown for different values of C. The probability shown is a time average over all GCR days.

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The growth of the GC

To account for the limited capacity of a GC during the competitive phase, we employed a variant of the stochastic logistic growth process [59], in which the death rate increases with the overall population size, from a basal rate of μ₀, as (6)

Here, N is the population capacity taken to be 200; n = (n₁, n₂,…,n_M) is the vector of cell numbers n_i for the M clones such that is the total number of B cells in the GC. The competitive phase lasts about 16 days in mice [31]. The total number of cells in the GC grows gradually until reaching the capacity N, where it remains approximately fixed.

Ag encounter and binding by the BCR

We assume here that an arm of a BCR at the tip of a B cell protrusion has a characteristic time τ to find an Ag molecule, after which the protrusion retracts empty. The probability that an arm of a BCR at the tip of a B cell protrusion finds Ag before time τ is (7) where is the on-rate for the BCR to find an Ag molecule given that there are N_Ag of them in its scanning radius, and the sequence-dependent on-rate with which a particular BCR binds to Ag is q. This leads to (8) where p₀ is the probability that no Ag is encountered, while p₁ is the probability that one of the arms encounters an Ag molecule in this time period.

Ag internalization

In order to capture the Ag, the BCR attached to a protrusion of the B cell applies a pulling force that works against the interactions of the BCR with Ag, and that between the Ag and the virus/Fc receptor/FDC membrane (see Fig 1). We denote the potential interaction energy required to extract the Ag by E_Ag−mem, and the interaction energy of the BCR/Ab and Ag by E_Ag−Ab.

The characteristic rupture time depends on the force applied by the B cell [60]. The force F does work x_bF on the bond, reducing its free energy and increasing the rupture rate to , where r₀ is the characteristic rate for bond disassociation when no force is present, x_b is the distance at which the bond ruptures, and β = k_BT. The intrinsic rupture rate with no force is estimated by Kramer’s escape from a potential barrier as (9) where K is a coefficient that depends on the shape of the interaction potential [61]. We take K = 1, and note that in writing Eq (9) we have assumed that the activation barrier to form the bond is much smaller than the energy gained upon binding (E_Ag−Ab). F is typically of the order of a few pico-Newtons [27]. We assume that each B cell has 100 BCRs but the qualitative behavior of our results does not depend on the number of BCRs. As each BCR can extract 0/1/2 Ag molecules, a B cell can extract between 0 and 200 Ag molecules.

Affinity change following BCR mutation

We modeled the effect of mutation as a change in the on-rate q or the interaction energy E_Ag−Ab upon cell division, with one daughter retaining the parent affinity, while for the other daughter: (10)

Here, N is a normal distribution with zero mean and standard deviation of , with D akin to an effective variability coefficient determining the magnitude of the change in q or E_Ag−Ab. Within this model, the energy, or q, can increase or decrease with equal probability at every division. We added a reflecting boundary condition at zero such that q is never negative. Since the off-rate scales as , the affinity of the BCR is calculated as ω = q/r₀.

Simulation algorithm

To study the time evolution of the GC reaction, we performed Brownian dynamics simulations. At every time point B cells proliferate or die with a probability which is determined by the time step and the proliferation and death rates (Table 2). We choose the time step to be smaller than the average time for proliferation. The simulation proceeds according to the following algorithm:

1. Initial seed of the GC with M clones.
2. Growth phase. B cells proliferate and die without competition according to a stochastic birth-death process with rates β₀, μ₀. This phase lasts T_growth days. The values of these parameters is detailed in Table 2.
3. Competitive phase. B cells proliferate and die with competition according to a stochastic birth-death process with birth rate (see Eq (1)) and logistic death rate (see Eq (6)). This phase lasts T_comp days. The values of these parameters is detailed in Table 2. In the stochastic simulation, at each time step:
   1. B cells attempt to capture Ag (see section “Ag encounter and binding by the BCR”, “Ag internalization”).
   2. B cells divide with the rate given by β_i (see Eq (1)). B cells have to capture at least 1 Ag molecule to attempt proliferation.
   3. The progeny of a B cell mutates its BCR affinity (see section “Affinity change following BCR mutation”).
   4. B cells die with a rate μ(n).

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Table 2. Simulation parameters.

https://doi.org/10.1371/journal.pcbi.1006408.t002